Dirac Operators for the Dunkl Angular Momentum Algebra

Symmetry Integrability and Geometry-Methods and Applications

article

Kieran Calvert¹  Marcelo De Martino² 

[1] Department of Mathematics, University of Manchester;Department of Electronics and Information Systems, University of Ghent

关键词: Dirac operators;    Calogero-Moser angular momentum;    rational Cherednik algebras.;   

DOI  :  10.3842/SIGMA.2022.040

来源: National Academy of Science of Ukraine

【 摘 要 】

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.

【 授权许可】

Unknown